First try flipping the breaker off/on slowly a couple times. Sometimes you can loose half the line without actually tripping the breaker and the heating coil requires the full 240 volts.
If this does nothing.
Measure the voltage at the plug
L1 to L2 should be 240 volts
L1 to Neutral and L2 to Neutral, both should be 120 volts.
If OK
Unplug the unit and check the wires at the unit's terminal strip to ensure they are properly connected and none of them have burned off
If OK
Plug the unit in and check the voltage at the terminal strip. This is just in case you have a bad line cord. Be careful 240 volts is lethal.
If OK
Unplug the unit and check the heating coil and thermostats etc. for continuity.

Odds are that the thermal fuse on the heater is blown.
Unplug the unit and check it with a meter, should be 0 ohms.
Be sure to unplug one side of it when checking it.

If it is blown you have to find out what caused it to go.
Note: that sometimes they do just blow on their own but changing it without checking other things is a gamble.

Check the heating coil.
Unplug the unit and both wires to the coil.
Check it with a meter, should be around 12 ohms.
Then check from each side of the coil to the case/frame, both should be infinite ohms (open). If not the coil may have sagged or broken and is touching the case. This can cause it to run on high and the thermostats cannot regulate it.

If the above is OK then you will also have to replace the hi-limit as it should have regulated the temperature so the fuse did not blow.

You still have to find out why it blew.
Check the seal (drum etc) in the unit. The air is pulled over the heating coils, through the drum and pushed out the exhaust. So any large seal leak will pull in room air and the cycling thermostat on the blower will run the unit hot.
Check that the lint filter is not coated with fabric softener residue which greatly reduces air flow.
Check/clean your vent system, I would do this even if the heating element is grounded.
Check/clean the blower wheel, would also do this.

If all OK you may want to replace the cycling thermostat as it's contacts may not be opening (welded shut).

If you do not own a meter, I would suggest you purchase a one. You can get a decent digital multimeter for under $20.00. You do not need fancy though it is nice if the leads are a couple feet long.
If it saves ordering one unnecessary part it has paid for itself and you end up owning a useful tool.
Most places will not let you return electrical parts so if you order it, you own it.
A couple things to watch when measuring ohms and continuity
1. Always remove power from the machine otherwise you could blow your meter.
2. Always disconnect at least one side of any device you are checking. This eliminates the possibility of measuring an alternate/parallel circuit path.
3. When checking for closed contacts and continuity use the lowest scale (Usually 200 ohms). Then try higher scales. This scale is 0 to 200 ohms so if the device you are measuring is 300 ohms this scale would show an open circuit which it is not, you are just measuring outside the scale's dynamic range.

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